1 Chapter 5 – The Time Value of MoneyCopyright 2008 John Wiley & Sons MT 480 Unit 2 CHAPTER 5 The...

1Chapter 5 – The Time Value of Money Copyright 2008 John Wiley & Sons

MT 480 Unit 2MT 480 Unit 2

CHAPTER 5The Time Value of MoneyThe Time Value of Money


How does a manager determine the value of a series of future cash flows, whether paying for an asset or evaluating a project?

We refer to this value as the time value of money (TVM).

The Time Value of MoneyThe Time Value of Money

What is the value of a stream of future cash flows today?


TVM is based on the belief that people prefer to consume goods today rather than wait to consume similar goods tomorrow. People have a positive time preference.


Consuming Today or Tomorrow

Money has a time value because a dollar today is worth more than a dollar tomorrow.


Today’s dollar can be invested to earn interest or spent.

Value of a dollar invested (positive interest rate) grows over time.

Rate of interest determines trade-off between spending today versus saving.


Consuming Today or Tomorrow



Future Value versus Present Value

Future value measures what one or more cash flows are worth at the end of a specified period.

Present value measures what one or more cash flows that are to be received in the future will be worth today (at t=0).

Financial decisions are evaluated either on a future value basis or present value basis.



Discounting is the process of converting future cash flows to their present values.

Compounding is the process of earning interest over time.

Future Value versus Present Value


Future Value and Future Value and Compounding Compounding

Single Period Investment

We can determine the value of an investment at the end of one period if we know the interest rate to be earned by the investment.

If you invest for one period at an interest rate of i, your investment, or principle, will grow by (1 + i) per dollar invested.

The term (1+ i) is the future value interest factor, often called simply the future value factor.



Two-Period Investing

After the first period, interest accrues on original investment (principle) and interest earned in preceding periods.

A two-period investment is simply two single-period investments back-to-back.


The principal is the amount of money on which interest is paid.

Simple interest is the amount of interest paid on the original principal amount only.

Compounding interest consists of both simple interest and interest-on-interest.


Two-Period Investing



General equation to find the future value after any number of periods.

The Future Value Equation

We can use financial calculators or future value tables to find the future value factor at different interest rates and maturity periods.

The term (1 + i)n is the future value factor.


where:

FVn = future value of investment at the end of period n

PV = original principle (P0) or present value

i = the rate of interest per period, which is often a year

n = the number of periods

(5.1) n

i1PVFV )( n

The general equation to find the future value is:



Compounding More Frequently Than Once a Year

The more frequently the interest payments are compounded, the larger the future value of $1 for a given time period.

where: m = number of compounding periods in a year


m×nnFV =PV×(1+i/m) (5.2)


When interest is compounded on a continuous basis, we can use the equation below.

where: e = exponential function which is about 2.71828


FV PV (5.3)i ne


Continuous compounding example


0.05 5

0.25

FV = $10,000

= $10,000 2.71828

= $10,000 1.284025

= $12,840.25

e

Your grandmother wants to put $10,000 in a savings account at a bank. How much money would she have at the end of five years if the bank paid 5 percent annual interest compounded continuously?


Present Value and Present Value and Discounting Discounting

Present value calculations state the current value of a dollar in the future.

This process is called discounting, and the interest rate i is known as the discount rate.

The present value (PV) is often called the discounted value of future cash payments.

The present value factor is more commonly called the discount factor.


The equation below gives us the general equation to find the present value after any number of periods.

(5.4) ni)(1nFV

PV




The further in the future a dollar will be received, the less it is worth today.

The higher the discount rate, the lower the present value of a dollar.


Finding the Interest RateFinding the Interest Rate

A number of situations will require you to determine the interest rate (or discount rate) for a given stream of future cash flows.

to determine the interest rate on a loan.

to determine a growth rate.

to determine the return on an investment.

For an individual investor or a firm, it may be necessary.


Compound Growth RatesCompound Growth Rates

Compound growth occurs when the initial value of a number increases or decreases each period by the factor (1 + growth rate).

(5.6) n

g)(1PVnFV

Examples include population growth, earnings growth.


CHAPTER 6CHAPTER 6

Discounted Cash Flows and ValuationDiscounted Cash Flows and Valuation

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Multiple Cash FlowsMultiple Cash Flows

Many business situations call for computing present value of a series of expected future cash flows. Determining market value of security. Deciding whether to make capital investment.

Process similar to determining future value of multiple cash flows.

Present Value of Multiple Cash Flows

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Next, calculate present value of each cash flow using equation 5.4 from the previous chapter.

Present Value of Multiple Cash Flows

Finally, add up all present values.

Sum of present values of stream of future cash flows is their current market price, or value.

First, prepare timeline to identify magnitude and timing of cash flows.

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Multiple Cash FlowsMultiple Cash Flows


Annuities and Perpetuities

Individual investors may make constant payments on home or car loans, or invest fixed amount year after year saving for retirement.

Many situations exist where businesses and individuals would face either receiving or paying constant amount for a length of period.

Level Cash FlowsLevel Cash Flows

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Annuity: any financial contract calling for equally spaced level cash flows over finite number of periods.

Annuities and Perpetuities

Perpetuity: contract calling for level cash flow payments to continue forever.

Ordinary annuities: constant cash flows occurring at end of each period.

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Present Value of an Annuity Can calculate present value of annuity same way

present value of multiple cash flows is calculated. Becomes tedious with large no. of payments.

Instead, simplify equation 5.4 in chapter 5 to obtain annuity factor. Results in equation 6.1 that can be used to

calculate the annuity’s present value.

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(6.1) i

i)(1

11

CF

i

factor) value Present(1CF

annuity an for factor value PresentCFPVA

n

n

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Finding Monthly or Yearly Payments Example

27


You have just purchased a $450,000 condominium. You were able to put $50,000 down and obtain a 30-year fixed rate mortgage at 6.125 percent for the balance. What are your monthly payments?

n 360

Monthly interest rate = 6.125 % / 12 months = 0.51042 %

1 1Present value factor = 0.1599589

(1+i) (1.0051042)

1 - Present value factorPV annuity factor =

i1 - 0.1599589

=

= 164.578400.0051042


Preparing a Loan Amortization Schedule

Amortization: the way the borrowed amount (principal) is paid down over life of loan.

Monthly loan payment is structured so each month portion of principal is paid off; at time loan matures, it is entirely paid off.

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Amortized loan: each loan payment contains some payment of principal and an interest payment.


Loan amortization schedule is a table showing: loan balance at beginning and end of each

period. payment made during that period. how much of payment represents interest. how much represents repayment of principal.

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With amortized loan, larger proportion of each month’s payment goes towards interest in early periods. As loan is paid down, greater proportion of

each payment is used to pay down principal.


Amortization schedules are best done on a spreadsheet.

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Finding the Interest Rate

The annuity equation can also be used to find interest rate or discount rate for an annuity.

To determine rate of return for the annuity, we need to solve equation for the unknown value i.

Other than using trial and error approach, easier to solve using financial calculator.

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Future Value of an Annuity

Future value annuity calculations usually involve finding what a savings or investment activity is worth at some future point.

E.g. saving periodically for vacation, car, house, or retirement.

We can derive the future value annuity equation from the present value annuity equation (equation 6.1). This results in equation 6.2.

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Future Value of an Annuity Equation

(6.2) i

1i)(1CF

i

1- factor value FutureCF

annuity an for factor value FutureCFFVA

n

n

33



Perpetuities A perpetuity is constant stream of cash flows that

goes on for infinite period.

In stock markets, preferred stock issues are considered to be perpetuities, with issuer paying a constant dividend to holders.

Equation for present value of a perpetuity can be derived from present value of an annuity equation with n tending to infinity.

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Important relationship between present value of annuity and a perpetuity.

Perpetuities

Just as perpetuity equation was derived from present value annuity equation, one can also derive present value of an annuity from the equation for a perpetuity.

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Annuity is called an annuity due when there is an annuity with payments being incurred at beginning of each period rather than at end.

Annuity Due

Rent or lease payments typically made at beginning of each period rather than at end.

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Annuity DueAnnuity transformation method shows relationship

between ordinary annuity and annuity due.

Each period’s cash flow thus earns extra period of interest compared to ordinary annuity. Present or future value of annuity due is

always higher than that of ordinary annuity.

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Annuity due = Ordinary annuity value (1+i) (6.4)


Annuity Due Example

The value of the annuity due shown in Exhibit 6.7B is:

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Annuity due = $3,312 (1.08) = $3,577


In addition to constant cash flow streams, one may have to deal with cash flows that grow at a constant rate over time.

These cash-flow streams called growing annuities or growing perpetuities.

Cash Flows That Grow at a Cash Flows That Grow at a Constant RateConstant Rate

39


Growing Annuity

Business may need to compute value of multiyear product or service contracts with cash flows that increase each year at constant rate.

These are called growing annuities.

Example of growing annuity: valuation of growing business whose cash flows increase every year at constant rate.

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Growing Annuity

Use this equation to value the present value of growing annuity (equation 6.5) when the growth rate is less than discount rate.

(6.5) i1

g11

g-i

CFPVA

n1

n

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Growing PerpetuityWhen cash flow stream features constant growing

annuity forever.

Can be derived from equation 6.5 when n tends to infinity and results in the following equation:

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1CFPVA = (6.6)

i - g


Interest rates can be quoted in financial markets in variety of ways.

Most common quote, especially for a loan, is annual percentage rate (APR).

APR represents simple interest accrued on loan or investment in a single period; annualized over a year by multiplying it by appropriate number of periods in a year.

Effective Annual Interest RateEffective Annual Interest Rate

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Calculating the Effective Annual Rate (EAR)Correct way to compute annualized rate is to

reflect compounding that occurs; involves calculating effective annual rate (EAR).

Effective annual interest rate (EAR) is defined as annual growth rate that takes compounding into account.

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Calculating the Effective Annual Rate (EAR)

EAR = (1 + Quoted rate/m)m – 1 (6.7)

m is the # of compounding periods during a year.

EAR conversion formula accounts for number of compounding periods, thus effectively adjusts annualized interest rate for time value of money.

EAR is the true cost of borrowing and lending.

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Consumer Protection Acts and Interest Rate DisclosuresTruth-in-Lending (1968) ensures that true cost of

credit was disclosed to consumers, so they could make sound financial decisions.

Truth-in-Savings Act provides consumers accurate estimate of return they would earn on investment.

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Consumer Protection Acts and Interest Rate Disclosures

Require that APR be disclosed on all consumer loans and savings plans, and prominently displayed on advertising and contractual documents.

Note that EAR, not APR, is the appropriate rate to use in present and future value calculations.

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Problem 5.18Problem 5.18


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